Risk Aversion - Expected Utility over Money

yedlu, Fall 2024

Notations

  • \(I = [\underline{m}, \overline{m}]\) be the interval of monetary prizes in \(\mathbb{R}\). Consider this as a continuous set of consequences.
  • \(p: I \to [0,1]\) be the simple lottery with finite support.
    • \(\{m \in I: p(m) > 0\}\): only possible finite monetary prizes are considered.
  • \(\mathcal{L}\) be the set of simple lotteries over \(I\).
    • The preference relation on monetary simple lotteries is defined as \(\succsim \subseteq \mathcal{L} \times \mathcal{L}\) such that \(p \succsim q \Leftrightarrow (p,q) \in \succsim \,\, \Leftrightarrow U(p) > U(q)\).
  • vNM preferences on \(\mathcal{L}\) are represented by an expected utility formed function:

\[U(p) = \sum_{m \in I} u(m) p(m)\]

Evaluating Lotteries

First-Order Stochastic Dominance (FOSD)

Settings: \(F_p\) is the cumulative probability distribution function (cdf) of lottery \(p\).
\(F_p(m) = \sum_{\alpha \leq m} p(\alpha)\) is the probability that one can receive a monetary prize less than equal to \(m\).

Definition
Lottery \(p\) first-order stochastically dominates lottery \(q\) if

  • \(p \neq q\);
  • \(F_p(m) \leq F_q(m)\) for all \(m \in \mathbb{R}\).

Intuition: lottery \(p\) have a lower chance receiving prizes lower than \(m\) compare to lottery \(q\). This implies that lottery \(p\) have a higher chance receiving prizes higher than \(m\) compare to lottery \(q\).

Simple Improvements

Lottery \(p\) is a simple improvement over lottery \(q\) if

  • there exists \(\alpha \in (0,1]\);
  • \(m > m'\);
  • there exists \(r \in \mathcal{L}\)

such that
\[\begin{aligned} p &= \alpha \delta_m + (1-\alpha) r \\ q &= \alpha \delta_{m'} + (1-\alpha) r \\ \end{aligned}\]

Intuition: we can consider both \(p\) and \(q\) as a mixture of a certain outcome monetary prize (\(m\) and \(m'\)), and a generic lottery \(r\).
This is a special case of first-order stochastic dominance.

Utility representation:

\[\begin{aligned} U(p) = \alpha u(m) + (1-\alpha) u(r)p(r) > \alpha u(m') + (1-\alpha) u(r)p(r) = U(q) \end{aligned}\]

Monotine vNM Preferences

Let \(\succsim \subseteq \mathcal{L} \times \mathcal{L}\) be a vNM prefernce. The following conditions are equivalent:

  1. If \(p\) FOSD \(q\), then \(p \succ q\).
  2. If \(p\) is a simple improvement over \(q\), then \(p \succ q\).
  3. Every linear \(U: \mathcal{L} \to \mathbb{R}\) representing \(\succsim\) has a strictly increasing Bernoulli index u.

When \(\succsim\) satisfies three conditions above, we call it a monotone vNM preference.

Certainty Equivalent

Let \(\succsim \subseteq \mathcal{L} \times \mathcal{L}\) be a vNM prefernce represented by an expected utility function with a continuous index \(u: I \to \mathbb{R}\). Then \(C: \mathcal{L} \to \mathbb{R}\) given by \(\delta_{C(p)} \sim p\) is a well-defined function. \(C(p)\) is the certainty equivalent of lottery \(p\).

Risk Aversion

Risk Averse Preferences

Let \(\overline{p} = \sum_{x \in I} p(x) x\) be the expected payoff of lottery. The preference \(\succsim\) is risk averse when \(\delta_{\overline{p}} \succsim p\).

Simple Mean Preserving Spread

\(q\) is a simple mean preserving spread of \(p\) when for some \(\alpha \in (0,1]\), some \(r, r' \in \mathcal{L}\) and \(\overline{r'} = m\).

\[\begin{aligned} p & = \alpha \delta_{m} + (1-\alpha) r \\ q & = \alpha r' + (1-\alpha) r \end{aligned}\]

Intuition: \(p\) and \(q\) yields the same expected payoff, but \(q\) has higher variance than of \(p\).

Risk Aversion Theorems

Let \(\succsim\) be a monotone vNM preference. The following are equivalent:

  1. \(\succsim\) is risk averse;
  2. If \(q\) is a mean preserving spread over \(p\) then \(p \succ q\);
  3. Every linear \(U: \mathcal{L} \to \mathbb{R}\) representing \(\succsim\) has a concave Bernoulli function.

If \(u\) is a concave bernoulli function, we would have:

\[1/2 u(a) + 1/2 u (b) \leq u(a/2 + b/2)\]

Levels of Risk Aversion

\(\succsim\) is more risk averse than \(\succsim'\) when \(\delta_m \succsim' p\) implies \(\delta_m \succsim p\).

Suppose \(u, v\) are continuous Bernoulli indices for monotone vNM preferences \(\succsim\) and \(\succsim'\). Then the following conditions are equivalent:

  1. \(\succsim\) is more risk averse than \(\succsim'\)
  2. there exists a concave, strictly increasing function \(f: u(\mathbb{R}) \to \mathbb{R}\) such that \(v(m) = f \circ u(m) = f(u(m))\).
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